#10045691

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4119 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18,23,21)(19,27,29,24)(20,25,30,31)(22,28,32,26)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,19,23,29)(18,24,21,27)(20,26,30,28)(22,25,32,31)(35,36)(39,42)(43,44)(47,48);(2,8)(4,9)(7,14)(10,15)(18,21)(20,30)(24,27)(26,28)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,20,23,30)(18,25,21,31)(19,28,29,26)(22,27,32,24)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21,23,18)(19,24,29,27)(20,31,30,25)(22,26,32,28)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,22)(18,26)(19,25)(20,27)(21,28)(23,32)(24,30)(29,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9952887);